Soup Servings

You have two soups, A and B, each starting with n mL. On every turn, one of the following four serving operations is chosen at random, each with probability 0.25 independent of all previous turns:

• Pour 100 mL from type A and 0 mL from type B.
• Pour 75 mL from type A and 25 mL from type B.
• Pour 50 mL from type A and 50 mL from type B.
• Pour 25 mL from type A and 75 mL from type B.

Note:

• There is no operation that pours 0 mL from A and 100 mL from B.
• The amounts from A and B are poured simultaneously during the turn.
• If an operation asks you to pour more than you have left of a soup, pour all that remains of that soup.

The process stops immediately after any turn in which one of the soups is used up.

Return the probability that A is used up before B, plus half the probability that both soups are used up in the same turn. Answers within 10^-5 of the actual answer will be accepted.